---
tag: 08WWTN738R
kind: concept
name: Explicit and implicit differential equations
msc: [34A09]
uses: [0P97V20RW7]
sources:
  - url: https://en.wikipedia.org/wiki/Ordinary_differential_equation
    locator: Definition
---

## Summary

An [[ordinary differential equation|0P97V20RW7]] of [[order|0XK0G19SK8]] $n$ is explicit when it has the form $y^{(n)} = f\left(x, y, y', \dots, y^{(n-1)}\right)$ for a given function $f$.
It is implicit when it has the general form $F\left(x, y, y', \dots, y^{(n)}\right) = 0$ and is not solved for $y^{(n)}$.
Each explicit equation is also implicit, with $F = y^{(n)} - f$.

## Remarks

The equation $y' = x^2 + y^2$ is explicit. The equation $(y')^2 + y^2 = 1$ is implicit.
If $\partial F / \partial y^{(n)}$ is not zero at a point, the implicit function theorem gives an explicit equation near that point.
The basic existence theorems, such as the [[Picard–Lindelöf theorem|0P6KRD5B7A]], are for explicit equations.

## Explanations

### L1 human

An explicit equation tells you the highest derivative directly, as a formula in the lower ones.
An implicit equation only gives a relation, and you may have to solve it for $y'$ first.
For $(y')^2 + y^2 = 1$ you get two explicit equations, $y' = \sqrt{1 - y^2}$ and $y' = -\sqrt{1 - y^2}$.